andconsideredinthe`Analyst。’Andherethequestionbetweenusis，whetherIhaverightlyrepresentedthesenseofthosewordsevanescantjamaugmentailla，inrenderingthem，``lettheincrementsvanish，’’i。e。lettheincrementsbenothing，orlettherebenoincrements？Thisyoudeny；but，asyourmanneris，insteadofgivingareasonyoudeclaim。I，onthecontrary，affirm，theincrementsmustbeunderstoodtobequitegone，andabsolutelynothingatall。Myreasonis，becausewithoutthatsuppositionyoucanneverbringthequantityorexpressiondownto，theverythingaimedatbysupposingtheevanescence。Saywhetherthisbenotthetruthofthecase？Whethertheformerexpressionisnottobereducedtothelatter？
Andwhetherthiscanpossiblybedonesolongasoisarealquantity？
Icannotindeedsayyouarescrupulousaboutyouraffirmations，andyetIbelievethatevenyouwillnotaffirmthis；itbeingmostevident，thattheproductoftworealquantitiesissomethingreal；andthatnothingrealcanberejectedeitheraccordingtotheofgeometry，oraccordingtoSirIsaac’sownPrinciples；forthetruthofwhichIappealtoallwhoknowanythingofthesematters。Further，byevanescentmusteitherbemeant，letthem（theincrements）vanishandbecomenothing，intheobvioussense，orletthembecomeinfinitelysmall。ButthatthislatterisnotSirIsaac’ssenseisevidentfromhisownwordsintheverysamepage，thatis，inthelastofhis`IntroductiontotheQuadratures，’whereheexpresslysaith，voluiostenderequodinmethodofluxionumnonopussitfigurasinfiniteparvasingeometriamintroducere。Uponthewhole，youseemtohaveconsideredthisaffairsoverysuperficiallyasgreatlytoconfirmmeintheopinionyouaresoangrywith，towit，thatSirIsaac’sfollowersaremuchmoreeagerinapplyinghismethodthanaccurateinexamininghisprinciples。Youraiseadustaboutevanescentaugments，whichmayperhapsamuseandamazeyourreader，butIammuchmistakenifiteverinstructsorenlightenshim。For，tocometothepoint，thoseevanescentaugmentseitherarerealquantities，ortheyarenot。Ifyousaytheyare；Idesiretoknowhowyougetridoftherejectaneousquantity？Ifyousaytheyarenot；youindeedgetridofthosequantitiesinthecompositionwhereoftheyarecoefficients；butthenyouareofthesameopinionwithme，whichopinionyouarepleasedtocall（p。58）``amostpalpable，inexcusable，andunpardonableblunder，’’
althoughitbeatruthmostpalpablyevident。
34。Nothing，Isay，canbeplainertoanyimpartialreaderthanthat，bytheevanescenceofaugmentsintheabove—citedpassage，SirIsaacmeanstheirbeingactuallyreducedtonothing。But，toputitoutofalldoubtthatthisisthetruth，andtoconvinceevenyou，whoshewsolittledispositiontobeconvinced，Idesireyoutolookintohis``AnalysisperAequationesInfinitas’’（p。20），where，inhispreparationfordemonstratingthefirstruleforthesquaringofsimplecurves，youwillfindthat，onaparalleloccasion，speakingofanaugmentwhichissupposedtovanish，heinterpretsthewordevanescerebyessenihil。Nothingcanbeplainerthanthis，whichatoncedestroysyourdefence。Andyet，plainasitis，Idespairofmakingyouacknowledgeit；
thoughIamsureyoufeelit，andthereaderifheusethhiseyesmustseeit。Thewordsevanesceresiveessenihildo（touseyourownexpression）stareusintheface。Lo！Thisiswhatyoucall（p。56）``sogreat，sounaccountable，sohorrid，sotrulyBoeotianablunder，’’thataccordingtoyou，itwasnotpossibleSirIsaacNewtoncouldbeguiltyofit。Forthefuture，Iadviseyoutobemoresparingofhardwords；since，asyouincautiouslydealthemabout，theymaychancetolightonyourfriendsaswellasyouradversaries。Asformypart，Ishallnotretaliate。Itissufficienttosayyouaremistaken。ButIcaneasilypardonyourmistakes。
Though，indeed，youtellme，onthisveryoccasion，thatImustexpectnoquarterfromSirIsaac’sfollowers。AndItellyouthatIneitherexpectnordesireany。Myaimistruth。MyreasonsIhavegiven。Confutethem，ifyoucan。Butthinknottooverbearmeeitherwithauthoritiesorharshwords。Thelatterwillrecoiluponyourselves。Theformer，inamatterofscience，areofnoweightwithindifferentreaders；and，asforbigots，Iamnotconcernedaboutwhattheysayorthink。
35。Inthenextplaceyouproceedtodeclaimuponthefollowingpassage，takenfromtheseventeenthsectionofthe`Analyst。’
``Consideringthevariousartsanddevicesusedbythegreatauthorofthefluxionarymethod；inhowmanylightsheplacethhisfluxions；andinwhatdifferentwayheattemptstodemonstratethesamepoint：onewouldbeinclinedtothinkhewashimselfsuspiciousofthejustnessofhisowndemonstrations。’’ThispassageyoucomplainofasveryhardusageofSirIsaacNewton。Youdeclaimcopiously，andendeavourtoshowthatplacingthesamepointinvariouslightsisofgreatusetoexplainit；whichyouillustratewithmuchrhetoric。Butthefaultofthatpassageisnotthehardusageitcontains：but，onthecontrary，thatitistoomodest，andnotsofullandexpressiveofmysenseasperhapsitshouldhavebeen。
WouldyoulikeitbetterifIshouldsay—``Thevariousinconsistentaccountswhichthisgreatauthorgivesofhismomentumsandhisfluxionsmayconvinceeveryintelligentreaderthathehadnoclearandsteadynotionsofthem，withoutwhichtherecanbenodemonstration？’’IownfranklythatIseenoclearnessorconsistenceinthem。Youtellme，indeed，inMiltonicverse，thatthefaultisinmyowneyes，``Sothickadropserenehasquench’dtheirorbs，Ordimsuffusionveil’d。’’Atthesametimeyouacknowledgeyourselfobligedforthosevariouslightswhichhaveenabledyoutounderstandhisdoctrine。Butasforme，whodonotunderstandit，youinsultme，saying：``ForGod’ssake，whatisityouareoffendedat，whodonotstillunderstandhim？’’MaynotIanswer，thatIamoffendedforthisveryreason—becauseIcannotunderstandhimormakesenseofwhathesays？YousaytomethatIamallinthedark。
Iacknowledgeit，andentreatyouwhoseesoclearlytohelpmeout。
36。YouSir，withthebrighteyes，bepleasedtotellme，whetherSirIsaac’smomentumbeafinitequantity，oraninfinitesimal，oramerelimit？Ifyousayafinitequantity：bepleasedtoreconcilethiswithwhathesaithinthescholiumofthesecondlemmaofthefirstsectionofthefirstbookofhisPrinciples：Caveintelligasquantitatesmagnitudinedeterminatas，sedcogitasemperdiminuendassinelimite。
Ifyousay，aninfinitesimal：reconcilethiswithwhatissaidinhis`IntroductiontotheQuadratures’：Voluiostenderequodinmethodofluxionumnonopussitfigurasinfiniteparvasingeometriamintroducere。Ifyoushouldsay，itisamerelimit；bepleasedtoreconcilethiswithwhatwefindinthefirstcaseofthesecondlemmainthesecondbookofhisPrinciples：UbidelateribusAetBdeerantmomentorumdimidia，&；c。，wherethemomentsaresupposedtobedivided。Ishouldbeverygladapersonofsuchaluminousintellectwouldbesogoodastoexplainwhetherbyfluxionswearetounderstandthenascentorevanescentquantitiesthemselves，ortheirmotions，ortheirvelocities，orsimplytheirproportions：and，havinginterpretedtheminwhatsenseyouwill，thatyouwouldthencondescendtoexplainthedoctrineofsecond，third，andfourthfluxions，andshewittobeconsistentwithcommonsenseifyoucan。Youseemtobeverysanguinewhenyouexpressyourselfinthefollowingterms：``Idoassureyou，Sir，frommyownexperience，andthatofmanyotherswhomIcouldname，thatthedoctrinemaybeclearlyconceivedanddistinctlycomprehended’’（p。
31）。Anditmaybeuncivilnottobelievewhatyousosolemnlyaffirm，fromyourownexperience。ButImustneedsownIshouldbebettersatisfiedofthis，if，insteadofentertaininguswithyourrhetoric，youwouldvouchsafetoreconcilethosedifficulties，andexplainthoseobscurepointsabovementioned。ifeitheryou，oranyoneofthosemanywhomyoucouldnamewillbutexplaintootherswhatyousoclearlyconceiveyourselves，Igiveyoumywordthatseveralwillbeobligedtoyouwho，Imayventuretosay，understandthosemattersnomorethanmyself。But，ifIamnotmistaken，youandyourfriendswillmodestlydeclinethistask。
37。Ihavelongagodonewhatyousooftenexhortmetodo—diligentlyreadandconsideredtheseveralaccountsofthisdoctrinegivenbythegreatauthorindifferentpartsofhiswritings；
anyuponthewholeIcouldnevermakeitouttobeconsistentandintelligible。
Iwasevenleadtosaythat``onewouldbeinclinedtothinkhewashimselfsuspiciousofthejustnessofhisowndemonstrations；andthathewasnotenoughpleasedwithanyonenotionsteadilytoadheretoit。’’AfterwhichIadded，``Thismuchisplain，thatheownedhimselfsatisfiedconcerningcertainpoints，whichneverthelesshecouldnotundertaketodemonstratetoothers。’’（Seetheseventeenthsectionofthe`Analyst。’）Itisonethingwhenadoctrineisplacedinvariouslights；andanotherwhentheprinciplesandnotionsareshifted。Whennewdevicesareintroducedandsubstitutedforothers，adoctrineinsteadofbeingillustratedmaybeexplainedaway。Whethertherebenotsomethingofthisinthepresentcase，Iappealtothewritingsofthegreatauthor—his`MethodusRationumPrimarumetUltimarum，’hissecondlemmainthesecondbookofhis`Principles，’
his`IntroductionandTreatiseoftheQuadratureofCurves。’Inallwhich，itappearstome，thereisnotoneuniformdoctrineexplainedandcarriedthroughoutthewhole，butrathersundryinconsistentaccountsofthisnewMethod，whichstillgrowsmoredarkandconfusedthemoreitishandled：
Icouldnothelpthinking，thegreatestgeniusmightlieundertheinfluenceoffalseprinciples；andwheretheobjectandnotionswereexceedinglyobscure，hemightpossiblydistrustevenhisowndemonstrations。``Atleastthusmuchseemedplain，thatSirIsaachadsometimesownedhimselfsatisfied，wherehecouldnotdemonstratetoothers。InproofwhereofImentionedhislettertoMr。Collins；hereuponyoutellme：thereisagreatdealofdifferencebetweensaying，Icannotundertaketoproveathing，andIwillnotundertakeit。’’But，inanswertothis，IdesireyouwillbepleasedtoconsiderthatIwasnotmakingapreciseextractoutofthatletter，inwhichtheverywordsofSirIsaacshouldalonebeinserted。
ButImademyownremarkandinferencefromwhatIrememberedtohavereadinthatletter；where，speakingofacertainmathematicalmatter，SirIsaacexpressethhimselfinthefollowingterms：``IsisplaintomebythefountainIdrawitfrom，thoughIwillnotundertaketoproveittoothers。’’Now，whethermyinferencemaynotbefairlydrawnfromthosewordsofSirIsaacNewton，andwhetherthedifferenceastothesensebesogreatbetweenwillandcaninthatparticularcase，Ileavetobedeterminedbythereader。
38。Inthenextparagraphyoutalkbigbutprovenothing。Youspeakofdrivingoutofintrenchments，ofsallying，andattacking，andcarryingbyassault；ofslightanduntenableworks，ofanew—raisedandundisciplinedmilitia，andofveteranregulartroops。Needthereaderbeamathematiciantoseethevanityofthisparagraph？Afterthisyouemploy（p。65）yourusualcolouring，andrepresentthegreatauthoroftheMethodofFluxions``asagoodoldgentlemanfastasleepandsnoringinhiseasychair；whileDameFortuneisbringinghimherapronfullofbeautifultheoremsandproblems，whichheneverknowsorthinksof。’’Thisyouwouldhavepassforaconsequenceofmynotions。ButIappealtoallthosewhoareeversolittleknowinginsuchmatters，whethertherearenotdiversfountainsofexperiment，induction，andanalogy，whenceamanmayderiveandsatisfyhimselfconcerningthetruthofmanypointsinmathematicsandmechanicalphilosophy，althoughtheproofsthereofaffordedbythemodernanalysisshouldnotamounttodemonstration？Ifurtherappealtotheconscienceofallthemostprofoundmathematicians，whethertheycan，withperfectacquiescenceofmind，freefromallscruple，applyanypropositionmerelyuponthestrengthofademonstrationinvolvingsecondorthirdfluxions，withouttheaidofanysuchexperiment，oranalogy，orcollateralproofwhatsoever？Lastly，Iappealtothereader’sownheart，whetherhecannotclearlyconceiveamediumbetweenbeingfastasleepanddemonstrating？
But，youwillhaveitthatIrepresentSirIsaac’sconclusionsascomingoutright，becauseoneerroriscompensatedbyanothercontraryandequalerror，whichperhapsheneverknewhimselfnorthoughtof：thatbyatwofoldmistakehearrivesthroughnotatscienceyetattruth：thatheproceedsblindfold，&；c。Allwhichisuntrulysaidbyyou，whohavemisappliedtoSirIsaacwhatwasintendedfortheMarquisdel’Hospitalandhisfollowers；
fornootherend（asIcansee）butthatyoumayhaveanopportunitytodrawthatingeniousportraitureofSirIsaacNewtonandDameFortune，aswillbemanifesttowhoeverreadsthe`Analyst。’
39。Youtellme（p。70）ifIthinkfittopersistinasserting``thatthisaffairofadoubleerrorisentirelyanewdiscoveryofmyown，whichSirIsaacandhisfollowersneverkneworthoughtof，thatyouhaveunquestionableevidencetoconvincemeofthecontrary，andthatallhisfollowersareclearlyapprisedthatthisveryobjectionofminewaslongsinceforeseen，andclearlyandfullyremovedbySirIsaacNewton，inthefirstsectionofthefirstbookofhis`Principia。’’’AllwhichIdoasstronglydenyasyouaffirm。AndIdoaverthatthisisanunquestionableproofofthematchlesscontemptwhichyou，Philalethes，havefortruth。AndIdoherepubliclycalluponyoutoproducethatevidencewhichyoupretendtohave，andtomakegoodthatfactwhichyousoconfidentlyaffirm。And，atthesametime，Idoassurethereaderthatyouneverwill，norcan。
40。IfyoudefendSirIsaac’snotions，asdeliveredinhis`Principia，’itmustbeontherigorousfootofrejectingnothing，neitheradmittingnorcastingawayinfinitelysmallquantities。IfyoudefendtheMarquis，whomyoualsostyleyourMaster，itmustbeonthefootofadmittingthatthereareinfinitesimals，thattheymayberejected，thattheyareneverthelessrealquantities，andthemselvesinfinitelysubdivisible。
Butyouseemtohavegrowngiddywithpassion，andintheheatofcontroversytohavemistakenandforgotyourpart。Ibeseechyou，Sir，toconsiderthattheMarquis（whomalone，andnotSirIsaac，thisdoubleerrorinfindingthesubtangentdothconcern）rejectsindeedinfinitesimals，butnotonthefootthatyoudo，towit，theirbeinginconsiderableinpracticalgeometryormixedmathematics。Butherejectsthemintheaccuracyofspeculativeknowledge：inwhichrespecttheremaybegreatlogicalerrors，althoughthereshouldbenosensiblemistakeinpractice；which，itseems，iswhatyoucannotcomprehend。Herejectsthemlikewiseinvirtueofapostulatum，whichIventuretocallrejectingthemwithoutceremony。And，thoughheinferrethaconclusionaccuratelytrue，yethedothit，contrarytotherulesoflogic，frominaccurateandfalsepremises。Andhowthiscomesabout，Ihaveatlargeexplainedinthe`Analyst，’andshewedinthatparticularcaseoftangents，thattherejectaneousquantitymighthavebeenafinitequantityofanygivenmagnitude，andyettheconclusionhavecomeoutexactlythesameway；and，consequently，thatthetruthofthismethoddothnotdependonthereasonassignedbytheMarquis，towit，thepostulatumforthrowingawayinfinitesimals；and，therefore，thatheandhisfollowersactedblindfold，asnotknowingthetruereasonfortheconclusionscomingoutaccuratelyright，whichIshewtohavebeentheeffectofadoubleerror。
41。Thisisthetruthofthematter，whichyoushamefullymisrepresentanddeclaimupon，tonosortofpurposebuttoamuseandmisleadyourreader。Forwhichconductofyoursthroughoutyourremarks，youwillpardonmeifIcannototherwiseaccount，thanfromasecrethopethatthereaderofyour`Defence’wouldneverreadthe`Analyst。’
Ifhedoth，hecannotbutseewhatanadmirablemethodyoutaketodefendyourcause：how，insteadofjustifyingthereasoning，thelogic，orthetheoryofthecasespecified，whichistherealpoint，youdiscourseofsensibleandpracticalerrors：andhowallthisisamanifestimpositionuponthereader。HemustneedsseethatIhaveexpresslysaid，``Ihavenocontroversyexceptonlyaboutyourlogicandmethod：thatIconsiderhowyoudemonstrate；whatobjectsyouareconversantabout；andwhetheryouconceivethemclearly。’’ThatIhaveoftenexpressedmyselftothesameeffect，desiringthereadertoremember，``thatIamonlyconcernedaboutthewayofcomingatyourtheorems，whetheritbelegitimateorillegitimate，clearorobscure，scientificortentative：thatIhave，onthisveryoccasion，topreventallpossibilityofmistake，repeatedandinsistedthatIconsiderthegeometricalanalystasalogician，i。e。sofarforthashereasonsandargues；andhismathematicalconclusions，notinthemselvesbutintheirpremises；notastrueorfalse，usefulorinsignificant，butasderivedfromsuchprinciples，andbysuchinferences。’’[`Analyst，’sect。20。]
Youaffirm（andindeedwhatcanyounotaffirm？）thatthedifferencebetweenthetruesubtangentandthatfoundwithoutanycompensationisabsolutelynothingatall。Iprofessmyselfofacontraryopinion。Myreasonis，becausenothingcannotbedividedintoparts。Butthisdifferenceiscapableofbeingdividedintoany，orintomorethananygivennumberofparts；forthetruthofwhichconsulttheMarquisdel’Hospital。And，betheerrorinfactorinpracticeeversosmall，itwillnotthencefollowthattheerrorinreasoning，whichiswhatIamaloneconcernedabout，isonewhittheless，itbeingevidentthatamanmayreasonmostabsurdlyabouttheminutestthings。
42。Prayanswermefairly，onceforall，whetheritbeyouropinionthatwhatsoeverislittleandinconsiderableenoughtoberejectedwithoutinconvenienceinpractice，thesamemayinlikemannerbesafelyrejectedandoverlookedintheoryanddemonstration。ifyousayNo，itwillthenfollowthatallyouhavebeensayinghereandelsewhere，aboutyards，andinches，anddecimalfractions，settingforthandinsistingontheextremesmallnessoftherejectaneousquantity，isquiteforeigntotheargument，andonlyapieceofskilltoimposeuponyourreader。IfyousayYes，itfollowsthatyouthengiveupatoncealltheordersoffluxionsandinfinitesimaldifferences；andsomostimprudentlyturnallyoursalliesandattacksandveteranstoyourownoverthrow。Ifthereaderisofmymind，hewilldespairofeverseeingyougetclearofthisdilemma。Thepointsincontroversyhavebeensooftenandsodistinctlynotedinthe`Analyst’thatIverymuchwonderhowyoucouldmistake，ifyouhadnomindtomistake。Itisveryplain，ifyouareinearnest，thatyouneitherunderstandmenotyourmasters。Andwhatshallwethinkofotherordinaryanalysts，whenitshallbefoundthatevenyou，wholikeachampionstepforthtodefendtheirprinciples，havenotconsideredthem？
43。Theimpartialreaderisentreatedtoremarkthroughoutyourwholeperformancehowconfidentyouareinasserting，andwithalhowmodestinprovingorexplaining：howfrequentitiswithyoutoemployfiguresandtropesinsteadofreasons：howmanydifficultiesproposedinthe`Analyst’arediscreetlyoverlookedbyyou，andwhatstrangeworkyoumakewiththerest：howgrosslyyoumistakeandmisrepresent，andhowlittleyoupractisetheadvicewhichyousoliberallybestow。Believeme，Sir，Ihadlongandmaturelyconsideredtheprinciplesofthemodernanalysis，beforeIventuredtopublishmythoughtsthereuponinthe`Analyst。’
And，sincethepublicationthereof，Ihavemyselffreelyconversedwithmathematiciansofallranks，andsomeoftheablestprofessors，aswellasmadeitmybusinesstobeinformedoftheopinionsofothers，beingverydesiroustohearwhatcouldbesaidtowardsclearingmydifficultiesoransweringmyobjections。But，thoughyouarenotafraidorashamedtorepresenttheanalystsasveryclearanduniformintheirconceptionofthesematters，yetIdosolemnlyaffirm（andseveralofthemselvesknowittobetrue）thatIfoundnoharmonyoragreementamongthem，butthereversethereof—thegreatestdissonance，andevencontrarietyofopinions，employedtoexplainwhatafterallseemedinexplicable。
44。Someflytoproportionsbetweennothings。Somerejectquantitiesbecauseinfinitesimal。Othersallowonlyfinitequantities，andrejectthembecauseinconsiderable。Othersplacethemethodoffluxionsonafootwiththatofexhaustions，andadmitnothingnewtherein。
Somemaintaintheclearconceptionoffluxions。Othersholdtheycandemonstrateaboutthingsincomprehensible。Somewouldprovethealgorismoffluxionsbyreductioadabsurdum，othersapriori。Someholdtheevanescentincrementstoberealquantities，sometobenothings，sometobelimits。
Asmanymen，somanyminds：eachdifferingonefromanother，andallfromSirIsaacNewton。Somepleadinaccurateexpressionsinthegreatauthor，wherebytheywoulddrawhimtospeaktheirsense；notconsideringthatifhemeantastheydo，hecouldnotwantwordstoexpresshismeaning。
Othersaremagisterialandpositive，saytheyaresatisfied，andthatisall；notconsideringthatwe，whodenySirIsaacNewton’sauthority，shallnotsubmittothatofhisdisciples。Someinsistthattheconclusionsaretrue，andthereforetheprinciples；notconsideringwhathathbeenlargelysaidinthe`Analyst’[Sect。19，20，&；c。]onthathead。Lastly，several（andthosenoneofthemeanest）franklyownedtheobjectionstobeunanswerable。
AllwhichImentionbywayofantidotetoyourfalsecolours：andthattheunprejudicedinquireraftertruthmayseeitisnotwithoutfoundationthatIcallonthecelebratedmathematiciansofthepresentagetoclearuptheseobscureanalytics，andconcuringivingtothepublicsomeconsistentandintelligibleaccountoftheirgreatMaster：foriftheydonot，Ibelievetheworldwilltakeitforgrantedthattheycannot。
45。HavinggonethroughyourdefenceoftheBritishmathematicians，Ifind，inthenextplace，thatyouattackmeonapointofmetaphysics，withwhatsuccessthereaderwilldetermine。Ihaduponanotheroccasionmanyyearsagowroteagainstabstractgeneralideas。
[Introductiontothe`TreatiseconcerningthePrinciplesofHumanKnowledge。’]
Inoppositiontowhich，youdeclareyourselftoadheretothevulgaropinion—thatneithergeometrynoranyothergeneralsciencecansubsistwithoutgeneralideas（p。74）。ThisimpliesthatIholdthattherearenogeneralideas。ButIholdthedirectcontrary—thatthereareindeedgeneralideas，butnotformedbyabstractioninthemannersetforthbyMr。Locke。Tomeitisplainthereisnoconsistentideaofthelikenesswhereofmaynotreallyexist：whatsoeverthereforeissaidtobesomewhatwhichcannotexist，theideathereofmustbeinconsistent。MrLockeacknowledgethitdothrequirepainsandskilltoformhisgeneralideaofatriangle。Hefartherexpresslysaithitmustbeneitherobliquenorrectangular，neitherequilateral，equicruralnorscalenum；butallandnoneoftheseoftheseatonce。Healsosaithitisanideawhereinsomepartsofseveraldifferentandinconsistentideasareputtogether。[`EssayonHumanUnderstanding，’
bk。iv，ch。vii，sect。9。]Allofwhichlooksverylikeacontradiction。
But，toputthematterpastdispute，itmustbenotedthatheaffirmsittobesomewhatimperfectthatcannotexist；consequentlytheideathereofisimpossibleorinconsistent。
46。Idesiretoknowwhetheritisnotimpossibleforanythingtoexistwhichdothnotincludeacontradiction：and，ifitis，whetherwemaynotinferthatwhatmaynotpossiblyexist，thesamedothincludeacontradiction：Ifurtherdesiretoknow，whetherthereadercanframeadistinctideaofanythingthatincludesacontradiction？Formypart，Icannot，norconsequentlyoftheabove—mentionedtriangle；thoughyou（youitseemsknowbetterthanmyselfwhatIcando）arepleasedtoassuremeofthecontrary。AgainIaskwhetherthatwhichitisabovethepowerofmantoformacompleteideaofmaynotbecalledincomprehensible？
Andwhetherthereadercanframeacompleteideaofthisimperfectimpossibletriangle？And，ifnot，whetheritdothnotfollowthatitisincomprehensible？
itshouldseemthatadistinctaggregateofafewconsistentpartswasnothingsodifficulttoconceiveorimpossibletoexist；andthat，therefore，yourcommentmustbewideoftheauthor’smeaning。Yougivemetounderstand（p82）thatthisaccountofageneraltrianglewasatrapwhichMr。Lockesettocatchfools。Whoiscaughtthereinletthereaderjudge。
47。ItisMr。Locke’sopinionthateverygeneralnamestandsforageneralabstractidea，whichprescindesfromthespeciesorindividualscomprehendedunderit。Thus，forexample，accordingtohim，thegeneralnamecolourstandsforanideawhichisneitherblue，red，green，noranyparticularcolour，butsomewhatdistinctandabstractedfromthemall。Tomeitseemsthewordcolourisonlyamoregeneralnameapplicabletoallandeachoftheparticularcolours：whiletheotherspecificnames，asblue，red，green，andthelike，areeachrestrainedtoamorelimitedsignification。Thesamecanbesaidofthewordtriangle。
Letthereaderjudgewhetherthisbenotthecase；andwhetherhecandistinctlyframesuchanideaofcolourasshallprescindfromallthespeciesthereof，orofatrianglewhichshallanswerMr。Locke’saccount，prescindingandabstractingfromalltheparticularsortsoftriangles，inthemanneraforesaid。
48。Ientreatmyreadertothink。For，ifhedothnot，hemaybeundersomeinfluencefromyourconfidentandpositivewayoftalking。Butanyonewhothinksmay，ifImistakenot，plainlyperceivethatyouaredeluded，asitoftenhappens，bymistakingthetermsforideas。
Nothingiseasierthantodefineintermsorwordsthatwhichisincomprehensibleinidea；forasmuchasanywordscanbeeitherseparatedorjoinedasyouplease，butideasalwayscannot。Itisaseasytosayaroundsquareasanoblongsquare，thoughtheformerbeinconceivable。Ifthereaderwillbuttakealittlecaretodistinguishbetweenthedefinitionandtheidea，betweenwordsorexpressionsandtheconceptionsofthemind，hewilljudgeofthetruthofwhatInowadvance，andclearlyperceivehowfaryouaremistakeninattemptingtoillustrateMr。Locke’sdoctrine，andwhereyourmistakelies。Or，ifthereaderismindertomakeashortwork，heneedsonlyatoncetotrywhether，layingasidethewords，hecanframeinhismindtheideaofanimpossibletriangle；uponwhichtrialtheissueofthisdisputemaybefairlyput。Thisdoctrineofabstractgeneralideasseemedtomeacapitalerror，productiveofnumberlessdifficultiesanddisputes，thatrunsnotonlythroughoutMr。Locke’sbook，butthroughmostpartsoflearning。Consequently，myanimadversionsthereuponwerenotaneffectofbeinginclinedtocarporcavilatasinglepassage，asyouwouldwrongfullyinsinuate，butproceededfromaloveoftruth，andadesiretobanish，sofarasinmelay，falseprinciplesandwrongwaysofthinking，withoutrespectofpersons。And，indeed，thoughyouandotherparty—menareviolentlyattachedtoyourrespectivemasters，yetI，whoprofessmyselfonlyattachedtotruth，seenoreasonwhyImaynotasfreelyanimadvertonMr。LockeorSirIsaacNewton，astheywouldonAristotleorDescartes。
Certainlythemoreextensivetheinfluenceofanyerror，andthegreatertheauthoritywhichsupportsit，themoreitdeservestobeconsideredanddetectedbysincereinquirersafterknowledge。
49。Inthecloseofyourperformance，youletmeunderstandthatyourzealfortruthandthereputationofyourmastershaveoccasionedyourreprehendingmewiththeutmostfreedom。Anditmustbeownedyouhaveshewnasingulartalenttherein。ButIamcomfortedundertheseverityofyourreprehensions，whenIconsidertheweaknessofyourarguments，which，weretheyasstrongasyourreproofs，couldleavenodoubtinthemindofthereaderconcerningthemattersindisputebetweenus。Asitis，Ileavehimtoreflectandexaminebyyourlight，howclearlyheisenabledtoconceiveafluxion，orafluxionofafluxion，apartinfinitelysmallsubdividedintoaninfinityofparts，anascentorevanescentincrement，thatwhichisneithersomethingnornothing，atriangleformedinapoint，velocitywithoutmotion，andtherestofthosearcanaofthemodernanalysis。Toconclude，Ihadsomethoughtsofadvisingyouhowtoconductyourselfforthefuture，inreturnfortheadviceyouhavesofreelyimpartedtome：but，asyouthinkitbecomesmerathertoinformmyselfthaninstructothers，Ishall，formyfartherinformation，takeleavetoproposeafewQueriestothoselearnedgentlemenofCambridge，whomyouassociatewithyourselfandrepresentasbeingequallysurprisedatthetendencyofmy`Analyst。’
50。Idesiretoknowwhetherthosewhocanneitherdemonstratenorconceivetheprinciplesofthemodernanalysis，andyetgiveintoit，maynotbejustlysaidtohaveFaith，andbestyledbelieversofMysteries？Whetheritisimpossibletofindamongthephysicians，mechanicalphilosophers，mathematicians，andphilomathematicians，ofthepresentage，somesuchbelievers，whoyetderideChristiansfortheirbeliefofmysteries？
Whetherwithsuchmenitisnotafair，reasonable，andlegitimatemethodtousetheargumentumadhominem？And，beingso，whetheritoughttosurpriseeitherChristiansorscholars？WhetherinanagewhereinsomanypretenderstoscienceattacktheChristianreligion，wemaynotbeallowedtomakereprisals，inordertoshewthattheirreligionofthosemenisnottobepresumedaneffectofdeepandjustthinking？Whetheranattempttodetectfalsereasonings，andremedydefectsinmathematics，oughttobeillreceivedbymathematicians？Whethertheintroducingmoreeasymethods，andmoreintelligibleprinciplesinanyscienceshouldbediscountenanced？Whethertheremaynotbefairobjectionsaswellascavils？
Andwhethertoinquirediligentlyintothemeaningoftermsandtheproofofpropositions，notexceptingagainstanythingwithoutassigningareason，noraffectingtomistakethesignificationofwords，orstickatanexpressionwherethesensewasclear，butconsideringthesubjectinalllights，sincerelyendeavouringtofindoutanysenseormeaningwhatsoever，candidlysettingforthwhatseemsobscureandwhatfallacious，andcallinguponthosewhoprofesstheknowledgeofsuchmatterstoexplainthem；whether，Isay，suchaproceedingcanbejustlycalledcavilling？Whethertherebeanipsedixiterected？Andifso，when，where，bywhom，anduponwhatauthority？
Whether，evenwhereauthoritywastotakeplace，onemightnothopethemathematics，atleast，wouldbeexcepted？Whetherthechiefend，inmakingmathematicssoconsiderableapartofacademicaleducation，benottoforminthemindsofyoungstudentshabitsofjustandexactreasoning？Andwhetherthestudyofabstruseandsubtlematterscanconducetothisend，unlesstheyarewellunderstood，examinedandsiftedtothebottom？Whether，therefore，thebringinggeometricaldemonstrationstotheseveresttestofreasonshouldbereckonedadiscouragementtothestudiesofanylearnedsociety？Whether，toseparatetheclearpartsofthingsfromtheobscure，todistinguishtherealprincipleswhereontruthsrest，andwhencetheyarederived，andtoproportionthejustmeasuresofassentaccordingtothevariousdegreesofevidence，beauselessorunworthyundertaking？
Whetherthemakingmoreofanargumentthanitwillbear，andplacingitinanunduerankofevidence，benotthelikelywaytodisparageit？Whetheritmaynotbeofsomeuse，toprovokeandstirupthelearnedprofessorstoexplainapartofmathematicallearningwhichisacknowledgedtobemostprofound，difficult，andobscure，andatthesametimesetforthbyPhilalethesandmanyothersasthegreatestinstancethathaseverbeengivenoftheextentofhumanabilities？Whether，forthesakeofagreatman’sdiscoveries，wemustadopthiserrors？Lastly，whetherinanagewhereinallotherprinciplesarecanvassedwiththeutmostfreedom，theprinciplesofFluxionsaretobealoneexcepted？AnAppendixconcerningMr。Walton’sVindicationofSirIsaacNewton’sPrinciplesofFluxions。1。Ihadnosoonerconsideredtheperformanceof`Philalethes，’
butMr。Walton’s`VindicationofFluxions’wasputintomyhands。AsthisDublinprofessorgleansafterthe`Cantabrigian，’onlyendeavouringtotranslateafewpassagesfromSirIsaacNewton’s`Principia，’andenlargeonahintortwoof`Philalethes，’hedeservesnoparticularnotice。Itmaysufficetoadvertisethereaderthattheforegoing`Defence’containsafullandexplicitanswertoMr。Walton，ashewillfind，ifhethinksitworthhispainstoreadwhatthisgentlemanhathwritten，andcompareittherewith：particularlywithsect。18，20，30，32—36，43。Itisnot，Iamsure，worthminetorepeatthesamethings，orconfutethesamenotionstwiceover，inmereregardtoawriterwhohathcopiedeventhemannersof`Philalethes，’andwhominansweringtheotherIhave，ifIamnotmuchmistakensufficientlyanswered。
2。Mr。Waltontouchesonthesamepointsthattheotherhadtoucheduponbeforehim。Hepursuesahintwhichtheotherhadgiven[`Philalethes，’p。32。]aboutSirIsaac’sfirstsectionconcerningtherationesprimaeetultimae。Hediscreetlyavoids，liketheother，tosayonesyllableofsecond，third，orfourthfluxions，andofdiversotherpointsmentionedinthe`Analyst，’aboutallwhichIobserveinhimamostprudentandprofoundsilence。Andyetheverymodestlygiveshisreadertounderstandthatheisabletoclearupalldifficultiesandobjectionsthathaveeverbeenmade（p。5）。Mr。Walton，inthebeginning，like`Philalethes，’
fromaparticularcasemakesageneralinference；supposingthatInfidelitytobeimputedtomathematiciansingeneralwhichIsupposeonlyinthepersontowhomthe`Analyst’wasaddressed，andcertainotherpersonsofthesamemindwithhim。Whetherthisextraordinarywayofreasoningbethecauseoreffectofhispassion，Iknownot：butbeforeIhadgottotheendofhis`Vindication，’Iceasedtobesurprisedathislogicandhistemperinthebeginning。Thedoubleerror，whichinthe`Analyst’wasplainlymeanttobelongtoothers，hewith`Philalethes’（whoseveryoversightheadopts）supposethtohavebeenascribedtoSirIsaacNewton（p。36）。
Andthiswriteralso，aswellasthe`Cantabrigian，’mustneedstakeuponhimtoexplainthemotiveofmywritingagainstfluxions；whichhegivesout，withgreatassurance，tohavebeenbecauseSirIsaacNewtonhadpresumedtointerposeinpropheciesandrevelations，andtodecideinreligiousaffairs（p。4）；whichissofarfrombeingtruethat，onthecontrary，Ihaveahighvalueforthoselearnedremainsofthatgreatman，whoseoriginalandfreegeniusisaneternalreproachtothattribeoffollowers，whoarealwaysimitatingbutneverresemblehim。ThisspecimenofMr。Walton’struthwillbeawarningtothereadertousehisowneyes，andinobscurepointsnevertotrustthegentleman’scandour，whodarestomisrepresenttheplainest。
3。Iwasthinkingtohavesaidnomoreconcerningthisauthor’sperformance，but，lestheshouldimaginehimselftoomuchneglected，Ientreatthereadertohavethepatiencetoperuseit；andifhefindsanyonepointinthedoctrineoffluxionsclearedup，oranyoneobjectioninthe`Analyst’answered，orsomuchasfairlystated，lethimthenmakehiscomplimentstotheauthor。But，ifhecannomoremakesenseofwhatthisgentlemanhaswrittenthanIcan，hewillneednoanswertoit。Nothingiseasierthanforamantotranslate，orcopy，orcomposeaplausiblediscourseofsomepagesintechnicalterms，wherebyheshallmakeashowofsayingsomewhat，althoughneitherthereadernorhimselfunderstandonetittleofit。WhetherthisbethecaseofMr。Walton，andwhetherheunderstandseitherSirIsaacNewton，orme，orhimself（whateverImaythink），Ishallnottakeituponmetosay。ButonethingIknow，thatmanyanunmeaningspeechpassethforsignificantbythemereassuranceofthespeaker，tillhecomethtobecatechizeduponit；andthenthetruthshowethitself。ThisVindicator，indeed，byhisdissemblingninepartsintenofthedifficultiesproposedinthe`Analyst，’shewethnoinclinationtobecatechizedbyme。Buthisscholarshavearighttobeinformed。I
thereforerecommendittothemnottobeimposedonbyhardwordsandmagisterialassertions，butcarefullytopryintohissense，andsifthismeaning，andparticularlytoinsistonadistinctanswertothefollowingQuestions。
4。Letthemaskhim，whetherhecanconceivevelocitywithoutmotion，ormotionwithoutextension，orextensionwithoutmagnitude？
Ifheanswersthathecan，lethimteachthemtodothesame。Ifhecannot，lethimbeasked，howhereconcilestheideaofafluxionwhichhegives（p。13）withcommonsense？—Again，lethimbeasked，Whethernothingbenottheproductofnothingmultipliedbysomething；and，ifso，whenthedifferencebetweenthegnomenandthesumoftherectangles[See`Vindication，’
p。17。]vanisheth，whethertherectanglesthemselvesdonotvanish？i。e。whenabisnothing，whetherAb+Babenotalsonothing？i。e。whetherthemomentumofABbenotnothing？—Lethimthenbeasked，whathismomentumsaregoodfor，whentheyarethusbroughttonothing？—Again，Iwishhewereaskedtoexplainthedifferencebetweenamagnitudeinfinitelysmallandamagnitudeinfinitelydiminished。Ifhesaith，thereisnodifference，thenlethimbefartherasked，howhedarestoexplainthemethodoffluxions，bytheratioofmagnitudesinfinitelydiminished（p。9），whenSirIsaacNewtonhathexpresslyexcludedallconsiderationofquantitiesinfinitelysmall？[Seehis`IntroductiontotheQuadratures。’]Ifthisablevindicatorshouldsaythatquantitiesinfinitelydiminishedarenothingatall，andconsequentlythat，accordingtohim，thefirstandlastratiosareproportionsbetweennothings，lethimbedesiredtomakesenseofthis，orexplainwhathemeansby``proportionbetweennothings，’’Ifheshouldsay，theultimateproportionsaretheratiosofmerelimits，thenlethimbeaskedhowthelimitsoflinescanbeproportionedordivided？Afterall，whoknowsbutthisgentleman，whohathalreadycomplainedofmeforanuncommonwayoftreatingmathematicsandmathematicians（p。5），may（aswellasthe`Cantabrigian’）cryout``SpainandtheInquisition！’’whenhefindshimselfthuscloselypursuedandbesetwithinterrogatories？Thatwemaynot，therefore，seemtoohardonaninnocentman，whoprobablymeantnothing，butwasbetrayedbyfollowinganotherintodifficultiesandstraitsthathewasnotawareof，Ishallproposeonesingleexpedient，bywhichhisdisciples（whomitmostconcerns）
maysoonsatisfythemselveswhetherthisVindicatorreallyunderstandswhathetakesuponhimtovindicate。Itis，inshort，thattheywouldaskhimtoexplainthesecond，third，orfourthfluxionsuponhisprinciples。
Bethisthetouchstoneofhis`Vindication。’Ifhecandoit，Ishallownmyselfmuchmistaken：ifhecannot，itwillbeevidentthathewasmuchmistakeninhimself，whenhepresumedtodefendfluxionswithoutsomuchasknowingwhattheyare。So，havingputthemeritsofthecauseonthisissue，Ileavehimtobetriedbyhisscholars。